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The bicategorical point of view provides a natural setting for many conceptsin the representation theory of monoidal categories. We show that centers oftwisted bimodule categories correspond to categories of 2-dimensional naturaltransformations and modifications between the deloopings of the twistingfunctors. We also show that dualities lift to centers of twisted bimodulecategories. Inspired by the notion of (pre)bimonoidal functors due to McCurdyand Street and by bilax functors of Aguiar and Mahajan, we study 2-dimensionalfunctors which are simultaneously lax and colax with a compatibility condition.Our approach uses a sort of 2-categorical Yang-Baxter operators, but the ideacould equally be carried out using a kind of 2-categorical braidings. We showhow this concept, which we call bilax functors, generalize many known notionsfrom the theory of Hopf algebras. We propose a 2-category of bilax functorswhose 1-cells generalize the notions of Yetter-Drinfel`d modules in ordinarycategories, and a type of bimonads and mixed distributive laws in 2-categories.We show that the 2-category of bilax functors from the trivial 2-category isisomorphic to the 2-category of bimonads, and that there is a faithful2-functor from the latter to the 2-category of mixed distributive laws of Powerand Watanabe.

Categorical centers and Yetter--Drinfel`d-modules as 2-categorical (bi)lax structures